3.2
Melting of ice particles
Recently, the modeling of dry snowflakes scattering properties using realistic snow particle shapes has received much attention from the scientific community. Considering the impor- tance of the melting layer in precipitation detection and quantification, the simulation of the scattering properties of partially melted snowflakes with a realistic spatial distribution of water should be the natural next step. However the computational cost of such investigation has greatly limited the number of studies focusing on this research field. The details in the microphysics of snow melting are also poorly understood, making difficult to design a model that adequately represents the melting process. There are only qualitative information about the melting process. A description on how to link the melted fraction to the distribution of water within the melting snowflake is still lacking. The melting model presented in this section uses all the available information on the melting microphysics and proposes a procedure to address the important problem of simulating the characteristics of the melting snowflakes.
The effect of spatial distribution of liquid water in mixed-phase particles on their scatter- ing properties has been investigated by Fabry and Szyrmer (1999). They defined six different models describing partially melted hydrometeors composed of aggregates of ice crystals and water. It is shown that the scattering model whose melting snow morphology resembles most the one of real snowflakes (discrete ice crystals covered by a thin layer of liquid water) reproduces the available radar observations at X-band with the highest accuracy.
Korkmaz (2004) exploited the Fabry and Szyrmer (1999) six models to simulate radar signatures at S, X, and Ka bands and compared the results with measured radar data. He found that the best performance is achieved for the model of aggregates with the largest ice density near the core. According to these results, among the various configurations considered, it is the model of ice aggregate crystals covered by a thin layer of liquid water that performs best in representing measured radar parameters of partially melted hydrometeors.
Observational studies (Knight, 1979) suggest that the water part of melting snow aggre- gates tend to fill the corners between ice crystals when they exists. Moreover, laboratory observations of melting snowflakes, performed by Oraltay and Hallett (2005), suggest that the melting process starts from the high-curvature regions of the surface of the crystals and capillary forces move water from high-curvature to low-curvature regions as melting continues.
Based on these results, the SAM model describes the melting process by starting from the ice phase aggregates derived in section 3.1. The dry and mixed-phase aggregates are described as clusters of polarizable regions belonging to a cubic lattice. In contrast to the aggregation phase of the SAM model the melting simulation is in fact based on the description
Fig. 3.6 Graphical representation of the regions of the cubic lattice that affects the melting probability of the central region.
of the particle in terms of volume elements and therefore it is not possible to increase the resolution of the lattice grid after the melting process.
A sequence of steps is performed to replace single ice lattice regions (points) with water ones. The aggregate mass and shape is maintained constant during the whole process allowing for a direct comparison of the scattering properties of the partially melted snowflakes with those of its dry counterpart.
Melting probability
For each ice lattice point a probability of melting is defined. The not normalized probability associated to each single ice dipole l in the lattice structure of coordinates (xl , yl, zl) is a
function of the physical state (type of material and its phase) of the six regions that share a surface of contact with it. A graphical representation of the regions of the cubic lattice affecting the melting probability of the central polarizable dipole is given in figure 3.6.
This probability of melting is described mathematically in compact way by Ptotl (xl, yl, zl) ∝
∑
j,k
Pstate(xl+ j Ik) i = −1, 1 j = 1, 2, 3 (3.8)
3.2 Melting of ice particles 49
Fig. 3.7 A two-dimensional representation of the technical procedure used to compute the melting probability. The color code for the type of material and phase of the lattice regions are as follows: blue for ice, red for water, and white for air. For each lattice region the melting probability is shown. As an example for the ice cell in the upper left corner the methodology that calculates the melting probability Ptot is specified.
Any lattice region might be composed of air, ice, or water (state index in equation 3.8). Only the six regions that have a non-null area of contact with the central region are accounted for in the computation of the melting probability of the central region in accordance with the Fourier’s law of heat conduction. The SAM algorithm was run setting Pair = 1, Pwater = 0.1,
and Pice= 0. A two-dimensional example of the computational scheme of the probability of
melting associated at each dipole region is given in figure 3.7.
The melting process algorithm statistically advantages the melting of the aggregate surface regions with respect to the inner parts of the particle since, at the initial steps, internal points, being surrounded only by other iced regions, have an associated probability to melt that is zero. The melting procedure is performed until the number of water regions reaches the value set by the user. The final configuration shows that the melted points are randomly distributed mostly on the particle surface (an example is given in figure 3.8).
The mixed-phase particle has the same mass and shape of the corresponding dry particle so that a direct comparison among the radar-derived parameters is possible. Since the natural
Fig. 3.8 Images of dry and partially melted aggregates snowflake with Dmax= 3 mm. The
same aggregate is shown with melted fraction equal to 0% (completely dry aggregate), 10%, 20% and 30%. The snowflakes are represented as a cluster of points with the lattice regions composed of ice colored with blue and the water ones are drawn in red.
3.2 Melting of ice particles 51
melting process changes the morphology of the snowflake, this method is not appropriate to model mixed-phase particles with large values of melted fraction. However, Figure 3.8 shows that 10% of melted fraction covers just a small part of the total particle surface; thus, the use of the same shape in the computations is an acceptable assumption. For larger values of melted fraction this assumption becomes questionable and the migration of the water mass to the inner parts of the snowflake should be taken into account.
Although the melting behavior is mostly guided by the snow crystal habit, observational studies have suggested the early stages of melting to be more intense on the lower part of the snowflake (Fujiyoshi, 1986; Mitra et al., 1990). This spatial enhanced melting has been described only qualitatively, and there is neither an indication for the early melting stage time extent nor for the melted mass fraction. Perhaps this differential melting between the upper and the lower part of the snowflake could increase the polarimetric signature of melting snowflakes and the study of this particular characteristic of snow melting is left for further studies.
Chapter 4
Scattering computations
The main purpose for snowflake shape modeling is to evaluate the sensitivity of snow particles’ single scattering properties to the non-homogeneous distribution of ice mass within the snowflake volume. In this chapter the single-scattering properties of the snowflakes modeled with the SAM algorithm (dry and mixed phased) are computed by using a discrete dipole approximation (DDA) code which allows to model irregularly shaped targets. In case of mixed-phased particles, realistic radiative properties are obtained by assuming snow aggregates with a 10% of melted fraction. The single-scattering properties are compared with those calculated through Mie theory together with Maxwell-Garnett effective medium approximation using both a homogeneous sphere and a layered-sphere models.
The single scattering properties analysis focus on the effects of internal mass radial distribution. It is known that due to the aggregation process not only the only snowflake density decreases with size, but it is also expected that density in the inner portion of snow aggregate is greater then the exterior part of it. This will likely have important effects on the scattering from a dry and melting snowflake: for example, since a greater proportion of the mass is confined in a small volume (considered for simplicity to be at the center of the snowflake), more of the melted mass will be closer to the center, resulting in smaller enhancements in the bright band than if it were distributed uniformly throughout the melting hydrometeor.
Some applications to quantitative precipitation estimation using radar data are presented to show how the resulting differences in the basic optical properties would propagate into radar measurable. Large discrepancies in the derivation of the equivalent water content and snowfall rate from radar measurements could be observed when large-size parameters are accounted for. The results of the present analysis has been published in an international peer reviewed scientific journal (Ori et al., 2014).